83 lines
2.1 KiB
Python
83 lines
2.1 KiB
Python
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"""
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=====================================
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Blind source separation using FastICA
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=====================================
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An example of estimating sources from noisy data.
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:ref:`ICA` is used to estimate sources given noisy measurements.
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Imagine 3 instruments playing simultaneously and 3 microphones
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recording the mixed signals. ICA is used to recover the sources
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ie. what is played by each instrument. Importantly, PCA fails
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at recovering our `instruments` since the related signals reflect
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non-Gaussian processes.
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"""
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# %%
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# Generate sample data
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# --------------------
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import numpy as np
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from scipy import signal
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np.random.seed(0)
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n_samples = 2000
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time = np.linspace(0, 8, n_samples)
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s1 = np.sin(2 * time) # Signal 1 : sinusoidal signal
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s2 = np.sign(np.sin(3 * time)) # Signal 2 : square signal
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s3 = signal.sawtooth(2 * np.pi * time) # Signal 3: saw tooth signal
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S = np.c_[s1, s2, s3]
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S += 0.2 * np.random.normal(size=S.shape) # Add noise
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S /= S.std(axis=0) # Standardize data
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# Mix data
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A = np.array([[1, 1, 1], [0.5, 2, 1.0], [1.5, 1.0, 2.0]]) # Mixing matrix
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X = np.dot(S, A.T) # Generate observations
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# %%
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# Fit ICA and PCA models
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# ----------------------
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from sklearn.decomposition import PCA, FastICA
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# Compute ICA
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ica = FastICA(n_components=3, whiten="arbitrary-variance")
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S_ = ica.fit_transform(X) # Reconstruct signals
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A_ = ica.mixing_ # Get estimated mixing matrix
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# We can `prove` that the ICA model applies by reverting the unmixing.
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assert np.allclose(X, np.dot(S_, A_.T) + ica.mean_)
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# For comparison, compute PCA
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pca = PCA(n_components=3)
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H = pca.fit_transform(X) # Reconstruct signals based on orthogonal components
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# %%
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# Plot results
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# ------------
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import matplotlib.pyplot as plt
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plt.figure()
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models = [X, S, S_, H]
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names = [
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"Observations (mixed signal)",
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"True Sources",
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"ICA recovered signals",
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"PCA recovered signals",
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]
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colors = ["red", "steelblue", "orange"]
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for ii, (model, name) in enumerate(zip(models, names), 1):
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plt.subplot(4, 1, ii)
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plt.title(name)
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for sig, color in zip(model.T, colors):
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plt.plot(sig, color=color)
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plt.tight_layout()
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plt.show()
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